In U.S. Pat. No. 4,073,005 tasks can be restricted to certain processors of the multiprocessor system and can be redispatched to run on other processors. This patent does not include a mechanism such as the signal dispatcher of the present invention which is invoked whenever a task element is put on the task dispatching queue to let other processors know that they are not processing the highest priority task and should invoke their task dispatcher to do a task switch. The present invention provides an advantage over the prior art in that it ensures that a processor of the multiprocessor system is processing the highest priority task.